Abstract
We show that the four-dimensional Chern-Simons theory studied by Costello, Witten and Yamazaki, is, with Nahm pole-type boundary conditions, dual to a boundary theory that is a three-dimensional analogue of Toda theory with a novel 3d W-algebra symmetry. By embedding four-dimensional Chern-Simons theory in a partial twist of the five-dimensional maximally supersymmetric Yang-Mills theory on a manifold with corners, we argue that this three-dimensional Toda theory is dual to a two-dimensional topological sigma model with A-branes on the moduli space of solutions to the Bogomolny equations. This furnishes a novel 3d-2d correspondence, which, among other mathematical implications, also reveals that modules of the 3d W-algebra are modules for the quantized algebra of certain holomorphic functions on the Bogomolny moduli space.
Highlights
Four-dimensional Chern-Simons theory studied by Costello, Witten and Yamazaki [4, 5] on I × S1 × Σ, with Nahm pole-type boundary conditions at the ends of the interval, I
We show that the four-dimensional Chern-Simons theory studied by Costello, Witten and Yamazaki, is, with Nahm pole-type boundary conditions, dual to a boundary theory that is a three-dimensional analogue of Toda theory with a novel 3d W-algebra symmetry
By embedding four-dimensional Chern-Simons theory in a partial twist of the five-dimensional maximally supersymmetric Yang-Mills theory on a manifold with corners, we argue that this three-dimensional Toda theory is dual to a two-dimensional topological sigma model with A-branes on the moduli space of solutions to the Bogomolny equations
Summary
The classical action of 5d maximally supersymmetric Yang-Mills (MSYM) theory is of the form [7, 8]. I.e., Mand M , take the meaning of spinor and R-symmetry indices, respectively This action is invariant under the supersymmetry transformations δAM = iζM M (ΓM )M NρN M (2.2a) δφM = ζM M. We shall take the underlying five-manifold to be of the form M = V × Σ, where submanifolds V and Σ correspond to the x1, x2, x3 and x4, x5 directions, respectively. Prior to carrying out the topological twist, we decompose the rotation group as SOM(5) → SOV (3) × SOΣ(2),. For t = ±i, one finds that the Q-transformations only depend on Σ via its complex structure Such a theory is topological on the submanifold V , but has holomorphic dependence on the remaining two directions along Σ. The boundary conditions we choose satisfy these requirements, as we shall explain in the subsection
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